1
vote
3answers
48 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
14
votes
2answers
508 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
1
vote
1answer
33 views

Evaluation of a Hankel-like determinant

I consider the following determinant (Hankel-like?) $$ [f_1,f_2,...,f_n]:=\begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ n-1 & f_1 & \cdots & f_{n-2}& f_{n-1}\\ 0 ...
2
votes
1answer
66 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
0
votes
2answers
38 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
0
votes
2answers
88 views

Reduce the expression $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}$ into a Geometric series [closed]

Is there a way to reduce the expression $$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}$$ into a Geometric series.
1
vote
4answers
45 views

Express $3.72444\ldots$ as a fraction using the formula for geometric progressions

I did the following: $$3.72\overline{4} = 3.724+\left(\left(\frac{4}{10^4}\right)+\left(\frac{4}{10^5}\right)+\left(\frac{4}{10^6}\right)+\cdots\right)$$ where $a=3.724$ and $r=\dfrac{1}{10}$ Using ...
-1
votes
3answers
76 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
2
votes
2answers
26 views

Algebraic Symbol Manipulation While Finding the Sum of a Series

In a precalc text, in the chapter on geometric progressions and series, we're told of the formula $S=\frac{a(1-r^{n+1})}{1-r}$ and that: $S=\frac{a}{1-r}$ is valid for $|r|<1$ We're then asked ...
1
vote
1answer
79 views

How to solve infinite series $\sum_{n=0}^\infty\frac{n}{2^{(n+1)}}$? [duplicate]

Can anyone please help me solve an infinite series: $$\sum_{n=0}^{\infty} \frac{n}{2^{(n+1)}}$$ I put it in Wolfram Alpha and got the result that it converges to $1$ I know that the infinite ...
1
vote
1answer
117 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
0
votes
2answers
31 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
3
votes
1answer
54 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
2
votes
1answer
56 views

Summing a Geometric Series

The 'bouncing ball' question shows up a number of times but I've not found this variation on it: In a section on geometric progressions we're asked to use the following formula: ...
6
votes
2answers
149 views

How to sum $\frac{1}{9} + \frac{1}{18}+\frac{1}{30}+\frac{1}{45} + …$

How to sum this series : $\frac{1}{9} + \frac{1}{18}+\frac{1}{30}+\frac{1}{45} + \frac{1}{65}......$ I am not getting any clue only a hint will be suffice please help. thanks..
4
votes
5answers
319 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
2
votes
1answer
18 views

an AP is changed to form a GP

Three numbers whose sum is $15$ are successive terms of an arithmetic series. If $1, 1$ and $4$ are added to these three numbers respectively, the resulting numbers are successive terms of a geometric ...
0
votes
4answers
30 views

Limit with log differentiation-help me understand why

Evaluate the following limit: $$\lim\limits_{x \to 0} (1+\arctan{(\frac{x}{2}}))^{\frac{2}{x}}$$ This is what I have-I know the answer is wrong, but I don't know why: ...
1
vote
1answer
63 views

How to compute $\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi -1)^{2n}}$?

I wrote as $\displaystyle\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi ...
1
vote
3answers
213 views

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\ $?

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\ $? I thought about it but I could not find a proof. Please give me some hints?
3
votes
2answers
122 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
1
vote
4answers
77 views

How is $2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$ calculated?

$$2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$$ I cant figure out why this is $\frac64$. I try to use telescopic series without success.
4
votes
5answers
492 views

Prove infinite series

$$ \frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2} $$ I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be ...
3
votes
4answers
94 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
-1
votes
2answers
25 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
2
votes
3answers
40 views

Confusion Over Sum of Geometric Series

On pg. 88 of A First Course in Probability, it says $$ P_i - P_1 = P_1[(q/p) + (q/p)^2 + \cdots + (q/p)^{i-1})] $$ Therefore: $$P_i = \frac{1 - (q/p)^i}{1 - q/p}P_1 $$ The series on the right in ...
3
votes
1answer
36 views

Generalisation of alternating functions

So if we want to have a function go positive negative we take $(-1)^n$, if we want it to take positive positive negative negative(like was on stack exchange a few days ago, we take: ...
3
votes
2answers
67 views

What is the sum of the power series below?

For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$ What is the sum of it?
1
vote
3answers
43 views

How to prove that $ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
1
vote
2answers
75 views

I don't know how to interpret this strange $\prod$

I have got a $\prod$ that is exactly as follows: $$\prod\limits_{k=0, k \ne k}^n \frac{x-c_k}{c_k-c_k}$$ I am not sure how to interpret this. My guesses are that it equals either $0, or ,1, or ...
4
votes
3answers
151 views

Series Question: $\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$$ I tried $$\frac{n+1}{2^nn^2}=\frac{1}{2^nn}+\frac{1}{2^nn^2}$$ The idea is using Riemann zeta function ...
6
votes
3answers
139 views

Series Question: $\sum_{n=1}^{\infty}\frac{1}{16n^2-1}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{16n^2-1}$$ I tried to use partial fraction ...
4
votes
4answers
124 views

Series Question: $\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$$ I tried to use partial fraction ...
-2
votes
1answer
63 views

Arithmetic and geometric progressions

An AP and GP with positive terms have the same number of terms and their first terms as well as last terms are equal . Show that the sum of the A.P. is greater than or equal to the sum of the GP.
1
vote
2answers
64 views

Recurrence relation: verify that the expression given for $a_n$

Consider the following difference equation and initial conditions. (i) $a_n=(-3)^n+n$, $a_0=1, a_1=-2$ satisfies (ii) $a_{n+1}+2a_n - 3a_{n-1} =4$ The answer is ...
3
votes
1answer
58 views

I can't figure out this Geometric Series

I'm in Precalc 2, and the question being asked is: "Show that the sum of the following infinite geometric series is $\dfrac{3}{2} = \dfrac{\sqrt{3}}{\sqrt{3} + 1} + \dfrac{\sqrt{3}}{\sqrt{3} + 3} ...
-1
votes
3answers
150 views

What's wrong with using algebra on infinite series?

I've recently found an article (referred somewhere on this site) criticizing the use of common rules of algebra on infinite series. To be honest, the video referred is one of the videos of Numberphile ...
2
votes
3answers
42 views

Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$

I'm looking for a closed-form expression (in terms of $n$), that will give the sequence $$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + ...
19
votes
12answers
711 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
0
votes
1answer
37 views

How does a sequence's convergency change finite sums?

What has been troubling me lately is that I cannot grasp how a finite series could ever diverge if a finite sequence that is divergent can only imply to a finite sum every time. Perhaps my main ...
1
vote
2answers
33 views

A problem regarding geometric progressions

Hello my homework included this problem and I really need a hint how to solve it. It says that the numbers $a_1,a_2 \ldots a_n$ form a geometric progression. Knowing $S=a_1+a_2+\ldots+a_n$ and $P=a_1 ...
1
vote
1answer
13 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
0
votes
2answers
22 views

Verification using Maclaurin series

The exercise asks to verify $$y(x) = x + \frac{2x^3}{3!} + \frac{24x^5}{5!} + ... = \frac{1}{2}\log\frac{1+x}{1-x}$$ by expanding $\displaystyle \frac{1}{2}\log\frac{1+x}{1-x}$ in Maclaurin's series ...
0
votes
1answer
24 views

Derive a formula to get the particular value from table

I have a table of points earned given the final game score. ...
1
vote
2answers
44 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
1
vote
4answers
111 views

Series question

I was trying this question: Consider the infinite series $$\frac{1}{1!} +\frac{4}{2!}+\frac{7}{3!}+\frac{10}{4!}+...$$ If the series continues with the same pattern, find the an expression for the ...
6
votes
1answer
45 views

Alternating Series and Convergence

The question is: Approximate the value of the series within an error of at most $10^{-4}$. $$ \sum_{n=1}^\infty \frac{(-1)^{(n+1)}}{(n+79)(n+73)} $$ According to $$|S_N - S| ≤ a_{N+1}$$ what is ...
2
votes
5answers
128 views

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing?

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing? My idea: $n^2<(n+1)^2 /+1$ $n^2+1<(n+1)^2+1/ \sqrt{}$ $\sqrt{n^2+1}<\sqrt{(n+1)^2+1}/+n$ ...
1
vote
2answers
57 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...